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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font> being   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>  st <font color="Olive" title="b1">T1</font>,<font color="Olive" title="b2">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a>  &amp; <font color="Olive" title="b1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Olive" title="b1">T1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/>( <font color="Olive" title="b2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Olive" title="b2">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Olive" title="b1">T1</font> );<br/>
<span class="kw">let </span><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a>  &amp; <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">i</font> = <span class="p1">(<span class="default"><a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="topdim_1.html#E14"><span class="lab"><font color="Green" title="E14">Lm3</font></span></a></span>;<br/>
<a NAME="E4:48"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">n1</font> = <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">T1</font>, <font color="Maroon" title="c7">T2</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">T1</font>,<font color="Maroon" title="c7">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a>  &amp; <font color="Maroon" title="c6">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:48_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c6">T1</font>,<font color="Maroon" title="c7">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 <span class="kw">and </span><br/><a NAME="E3:48_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c6">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 <span class="kw">and </span><br/><a NAME="E4:48_1"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">T1</font>,<font color="Maroon" title="c7">T2</font><span class="kw"> such that </span><br/><a NAME="E6:48_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">f9</font> = <font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> ;<br/>
<a NAME="E7:48_1"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E8:48_1"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E9:48_1"/><span class="lab"><font color="Green" title="E40">B1</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E10:48_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">T1</font> or  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">T1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">T1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )</span><br/><div class="add"><a NAME="E2:48_1_1_1"/><span class="kw">then </span>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_1"/>
( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E1:48_1_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="topdim_1.html#D4" target="_self" title="TOPDIM_1:def.4">Def4</a>, <a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">T1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )</span><br/><div class="add"><a NAME="E2:48_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c6">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> =  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a></span>;<br/><a NAME="E4:48_1_1_2"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 not <font color="Maroon" title="c6">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a></span>;<br/><a NAME="E5:48_1_1_2"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 not <font color="Maroon" title="c7">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E1:48_1_1_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_2_1"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_1_2_1_1"/>
( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_1_2"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )</span><br/><div class="add"><a NAME="E2:48_1_1_2_1_2"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E5:48_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="tops_2.html#T49" title="TOPS_2:th.49">TOPS_2:49</a></span>;<br/><a NAME="E3:48_1_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">T2</font><br/>  for <font color="Olive" title="b2">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">T2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font>  st <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font> st <br/>( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">U</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">U</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font> st <br/>( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:48_1_1_2_1_2_1"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font> st <br/>( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<a NAME="E3:48_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_1_2"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E4:48_1_1_2_1_2_1"/>
( <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c13">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">U</font> &amp; <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c13">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E4:48_1_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:48_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="topgrp_1.html#T26" title="TOPGRP_1:th.26">TOPGRP_1:26</a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">T1</font><span class="kw"> such that </span><br/><a NAME="E6:48_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W</font>
 <span class="kw">and </span><br/><a NAME="E7:48_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon" title="c14">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c13">U</font>
 <span class="kw">and </span><br/><a NAME="E8:48_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 <span class="kw">and </span><br/><a NAME="E9:48_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E1:48_1_1_2_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E3:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="topdim_1.html#T16" target="_self" title="TOPDIM_1:th.16">Th16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">FW</font> =  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font>;<br/>
<a NAME="E10:48_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="topdim_1.html#E28"><span class="lab"><font color="Green" title="E22">Lm5</font></span></a></span>;<br/>
<a NAME="E11:48_1_1_2_1_2_1"/>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font>,<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="metrizts.html#T3" title="METRIZTS:th.3">METRIZTS:3</a></span>;<br/>
<a NAME="E12:48_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<font color="Maroon" title="c6">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span>,<font color="Maroon" title="c7">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</a></span>;<br/>
<a NAME="E13:48_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E8:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E9:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E10:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a></span>;<br/>
<a NAME="E14:48_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T18" target="_self" title="TOPDIM_1:th.18">Th18</a></span>;<br/>
<a NAME="E15:48_1_1_2_1_2_1"/>
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1"><span class="lab"><font color="Green" title="E40">B1</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>;<br/>
<a NAME="E16:48_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A27</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E5:48_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<a NAME="E17:48_1_1_2_1_2_1"/>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E8:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E9:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E10:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="txt" href="#E12:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E54">A25</font></span></a></span>;<br/>
<a NAME="E18:48_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A28</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E10:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">W9</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E4:48_1_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:48_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">W9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W9</font> &amp; <font color="Maroon" title="c16">W9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<a NAME="E20:48_1_1_2_1_2_1"/>
<font color="Maroon" title="c16">W9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c13">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="relat_1.html#T123" title="RELAT_1:th.123">RELAT_1:123</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:48_1_1_2_1_2_1"/>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">W9</font> &amp; <font color="Maroon" title="c16">W9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">U</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E8:48_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E16:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c14">W</font></span>)</span> = 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">W</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c14">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T42" title="TOPS_1:th.42">TOPS_1:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">W</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c16">W9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="funct_1.html#T64" title="FUNCT_1:th.64">FUNCT_1:64</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">W9</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c16">W9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E5:48_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="tops_2.html#T60" title="TOPS_2:th.60">TOPS_2:60</a></span>
<br/>.= 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T42" title="TOPS_1:th.42">TOPS_1:42</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E23:48_1_1_2_1_2_1"/>
(  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c16">W9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E18:48_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="txt" href="topdim_1.html#E28"><span class="lab"><font color="Green" title="E22">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:48_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T15" target="_self" title="TOPDIM_1:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_2_1_2"/>
( <font color="Maroon" title="c7">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c7">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c6">T1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E3:48_1_1_2_1_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="topdim_1.html#T16" target="_self" title="TOPDIM_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:48"/><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">T1</font>, <font color="Maroon" title="c5">T2</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">T1</font>,<font color="Maroon" title="c5">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a>  &amp; <font color="Maroon" title="c4">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c5">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_2"/><span class="lab"><font color="Green" title="E34">A30</font></span>: 
<font color="Maroon" title="c4">T1</font>,<font color="Maroon" title="c5">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 <span class="kw">and </span><br/><a NAME="E2:48_2"/><span class="lab"><font color="Green" title="E35">A31</font></span>: 
<font color="Maroon" title="c4">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 <span class="kw">and </span><br/><a NAME="E3:48_2"/><span class="lab"><font color="Green" title="E36">A32</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c5">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">T1</font>,<font color="Maroon" title="c5">T2</font><span class="kw"> such that </span><br/><a NAME="E5:48_2"/><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f9</font> = <font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> ;<br/>
<a NAME="E6:48_2"/><span class="lab"><font color="Green" title="E38">A34</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E7:48_2"/><span class="lab"><font color="Green" title="E39">A35</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">T2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E8:48_2"/><span class="lab"><font color="Green" title="E40">B1</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E9:48_2"/><span class="lab"><font color="Green" title="E41">A36</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_2_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c4">T1</font> or  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c4">T1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_2_1_1"/><span class="lab"><font color="Green" title="E42">A37</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c4">T1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c5">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> )</span><br/><div class="add"><a NAME="E2:48_2_1_1"/><span class="kw">then </span>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_2_1_1"/>
( <font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c5">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E1:48_2_1_1"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="ref" href="topdim_1.html#D4" target="_self" title="TOPDIM_1:def.4">Def4</a>, <a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_2_1_2"/><span class="lab"><font color="Green" title="E42">A38</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c4">T1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c5">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> )</span><br/><div class="add"><a NAME="E2:48_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">i</font> =  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a></span>;<br/><a NAME="E4:48_2_1_2"/><span class="lab"><font color="Green" title="E43">A39</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a></span>;<br/><a NAME="E5:48_2_1_2"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
 not <font color="Maroon" title="c4">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2_1_2"><span class="lab"><font color="Green" title="E42">A38</font></span></a></span>;<br/><a NAME="E6:48_2_1_2"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
 not <font color="Maroon" title="c5">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_2"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E1:48_2_1_2"><span class="lab"><font color="Green" title="E42">A38</font></span></a></span>;<br/><a NAME="E7:48_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A42</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E9:48_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="tops_2.html#T49" title="TOPS_2:th.49">TOPS_2:49</a></span>;<br/><a NAME="E8:48_2_1_2"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T2</font><br/>  for <font color="Olive" title="b2">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c6">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font>  st <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font> st <br/>( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">U</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">U</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font> st <br/>( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:48_2_1_2_1"/>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font> st <br/>( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<a NAME="E3:48_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2_1_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E4:48_2_1_2_1"/>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_2"><span class="lab"><font color="Green" title="E40">B1</font></span></a>, <a class="txt" href="#E9:48_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>;<br/>
<a NAME="E5:48_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E9:48_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E6:48_2_1_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<a NAME="E6:48_2_1_2_1"/>
( <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">U</font> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E9:48_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E5:48_2_1_2"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="txt" href="#E6:48_2_1_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="topgrp_1.html#T26" title="TOPGRP_1:th.26">TOPGRP_1:26</a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T1</font><span class="kw"> such that </span><br/><a NAME="E8:48_2_1_2_1"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">W</font>
 <span class="kw">and </span><br/><a NAME="E9:48_2_1_2_1"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c12">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">U</font>
 <span class="kw">and </span><br/><a NAME="E10:48_2_1_2_1"/><span class="lab"><font color="Green" title="E52">A48</font></span>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c12">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 <span class="kw">and </span><br/><a NAME="E11:48_2_1_2_1"/><span class="lab"><font color="Green" title="E53">A49</font></span>: 
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="txt" href="#E3:48_2_1_2_1"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="ref" href="topdim_1.html#T16" target="_self" title="TOPDIM_1:th.16">Th16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">W9</font> = <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">W</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E5:48_2_1_2"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="txt" href="#E6:48_2_1_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">W9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W9</font> &amp; <font color="Maroon" title="c13">W9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<a NAME="E13:48_2_1_2_1"/>
<font color="Maroon" title="c13">W9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_2_1_2_1"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="ref" href="relat_1.html#T123" title="RELAT_1:th.123">RELAT_1:123</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:48_2_1_2_1"/>
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W9</font> &amp; <font color="Maroon" title="c13">W9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">U</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_2"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E5:48_2_1_2_1"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="txt" href="#E8:48_2_1_2_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )</span><br/>

<a NAME="E15:48_2_1_2_1"/>
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_2_1_2_1"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="topdim_1.html#T5" target="_self" title="TOPDIM_1:th.5">Th5</a></span>;<br/>
<a NAME="E16:48_2_1_2_1"/><span class="kw">then </span>
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c12">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_2_1_2_1"><span class="lab"><font color="Green" title="E53">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E17:48_2_1_2_1"/><span class="kw">then </span>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c12">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c4">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_2_1_2_1"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>;<br/>
<a NAME="E18:48_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">W</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#T14" title="TOPGEN_1:th.14">TOPGEN_1:14</a></span>;<br/>
<a NAME="E19:48_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">W9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E6:48_2_1_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></span>;<br/>
<a NAME="E20:48_2_1_2_1"/><span class="kw">then </span>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c5">T2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#T14" title="TOPGEN_1:th.14">TOPGEN_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:48_2_1_2_1"/>
(  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c13">W9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:48_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T15" target="_self" title="TOPDIM_1:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E10:48_2_1_2"/>
( <font color="Maroon" title="c5">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c5">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c4">T1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_2_1_2"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="txt" href="#E8:48_2_1_2"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="ref" href="topdim_1.html#T16" target="_self" title="TOPDIM_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:48"/><span class="lab"><font color="Green" title="E34">A50</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E32">A3</font></span></a>);<br/></span>
<a NAME="E7:48"/>
<span class="p1">(<span class="default"><a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:48"/>
( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E34">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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